While spherical data arises in many contexts, including in directional statistics, the current tools for density estimation and population comparison on spheres are quite limited. Popular approaches for comparing populations (on Euclidean domains) mostly involvea two-step procedure: (1) estimate probability density functions (pdfs) from their respective samples, most commonly using the kernel density estimator, and, (2) compare pdfs using a metric such as the L2 norm. However, both the estimated pdfs and their differences depend heavily on the chosen kernels, bandwidths, and sample sizes. Here we develop a framework for comparing spherical populations that is robust to these choices. Essentially, we characterize pdfs on spherical domains by quantifying their smoothness. Our framework uses a spectral representation, with densities represented by their coefficients with respect to the eigenfunctions of the Laplacian operator on a sphere. The change in smoothness, akin to using different kernel bandwidths, is controlled by exponential decays in coefficient values. Then we derive a proper distance for comparing pdf coefficients while equalizing smoothness levels, negating influences of sample size and bandwidth. This signifies a fair and meaningful comparisons of populations, despite vastly different sample sizes, and leads to a robust and improved performance. We demonstrate this framework using examples of variables on S1 and S2, and evaluate its performance using a number of simulations and real data experiments.
A tree-based kernel for densities and its applications in clustering DNase-seq profiles
Department of Statistics & Data Science Institute, University of Chicago（芝加哥大学统计与数据科学研究所）；Department of Human Genetics, University of Chicago（芝加哥大学人类遗传学系）
Data-adaptive estimation of time-varying spectral densities
本文介绍了一种数据自适应非参数方法，用于从非平稳时间序列中估计时变谱密度。时变谱密度通常通过局部核平滑来估计。然而，这些非参数估计的性能，关键取决于平滑带宽，需要指定在时间和频率方向。作为传统带宽选择方法的替代和扩展，我们提出了一种自适应构造局部平滑核的迭代算法。受传播分离概念（Polzehl and Spokoiny 2006）的启发，其主要思想是确定时频平面中的一个点的最大局部邻域，在该邻域上，平滑是由数据证明的。通过对平滑核进行非参数化的整形，我们的方法不仅避免了严格意义上的带宽选择问题，而且变得更加灵活。它不仅适应于平滑变化的光谱中的曲率变化，而且还适应于时变光谱中的结构突变。
